Let $p$ be an odd prime number and $K_{\infty}$ the cyclotomic $\mathbb{Z}_{p}$-extension of a Galois $p$-extension $K$ over an imaginary quadratic field. We consider the Galois group $\tilde{X}(K_{\infty})$ of the maximal unramified pro-$p$-extension of $K_{\infty}$. In this paper, under certain assumptions, we give certain $K$ such that $\tilde{X}(K_{\infty})$ is abelian. Also, we give an example such that a special value of the characteristic polynomial of the Iwasawa module of $K_{\infty}$ determines whether $\tilde{X}(K_{\infty})$ is abelian or not.

The aim of this paper is to derive energy estimates for solutions of the Cauchy problem for the Klein--Gordon type equation $u_{tt} - \bigtriangleup u + m(t)^{2} u = 0$. The coefficient $m$ is given by $m(t)^{2} = \lambda(t)^{2} + p(t)$ with a decreasing, smooth shape function $\lambda$ and an oscillating, smooth and bounded perturbation function $p$. We study under which assumptions for $\lambda$ and $p$ one can expect results about a generalization of energy conservation. The main theorems of this note deal with $m$ belonging to $C^{M}$, $M \ge 2$, and $m$ belonging to the Gevrey class $\gamma^{(s)}$, $s \ge 1$.

Let $P$ be a finite metacyclic $2$-group and $\mathcal{F}$ a fusion system on $P$. We prove that $\mathcal{F}$ is nilpotent unless $P$ has maximal class or $P$ is homocyclic, i.e. $P$ is a direct product of two isomorphic cyclic groups. As a consequence we obtain the numerical invariants for $2$-blocks with metacyclic defect groups. This paper is a part of the author's PhD thesis.

Consider the Cauchy problem for a system of weakly coupled heat equations, whose typical one is \begin{equation*} \left\{ \begin{array}{@{}ll@{}} u_{t} - \Delta u = \lvert v \rvert^{p-1}v,\\ v_{t} - \Delta v = \lvert u \rvert^{q-1}u, & (t, x) \in \mathbf{R}_{+} \times \mathbf{R}^{N}, \end{array} \right. \end{equation*} with $p,q \ge 1$, $pq > 1$. When $p,q$ satisfy $\max((p+1)/(pq-1),(q+1)/(pq-1)) < N/2$, the exponents $p,q$ are supercritical. In this paper we assort the supercritical exponent case to two cases. In one case both $p$ and $q$ are bigger than the Fujita exponent $\rho_{F}(N)=1+2/N$, while in the other case $\rho_{F}(N)$ is between $p$ and $q$. In both cases we obtain the time-global and unique existence of solutions for small data and their asymptotic behaviors. These observation will be applied to the corresponding system of the damped wave equations in low dimensional space.

In this paper, Kra's distance $d_{K}$ and the hyperbolic distance $d_{\mathbb{D}}$ are compared on the unit disk $\mathbb{D}$. It is shown that $2d_{K} < d_{\mathbb{D}} < (\pi^{2}/8)\exp{d_{K}}$ on $\mathbb{D} \times \mathbb{D} \setminus \{\text{diagonal}\}$, where the constants $2$ and $\pi^{2}/8$ are sharp. As a consequence, this result gives a negative answer to a question posed by Martin [7] in a stronger sense.

In this paper we study geometric properties of the slowness surface of the system of crystal acoustics for cubic crystals in the special case when the stiffness constants satisfy the condition $a = -2b$. The paper is a natural continuation of the paper [9] in which related properties were studied for general constants $a$ and $b$, but assuming that we were in the nearly isotropic case, in which case $a - b$ has to be small. We also take this opportunity to correct a statement made in [9]: see Remark 1.3.

Let $f(x)$ be a monic polynomial in $\mathbb{Z}[x]$. We have observed a statistical relation of roots of $f(x) \bmod p$ for different primes $p$, where $f(x)$ decomposes completely modulo $p$. We could guess what happens if $f(x)$ is irreducible and has at most one decomposition $f(x) = g(h(x))$ such that $g,h$ are monic polynomials over $\mathbb{Z}$ with $h(0) = 0$, $1 < \deg h < \deg f$. In this paper, we study cases that $f$ has two different such decompositions. Besides, we construct a series of polynomials f which have two non-trivial different decompositions $f(x) = g(h(x))$.

We prove pathwise uniqueness for stochastic differential equations driven by non-degenerate symmetric $\alpha$-stable Lévy processes with values in $\mathbb{R}^{d}$ having a bounded and $\beta$-Hölder continuous drift term. We assume $\beta > 1 - \alpha/2$ and $\alpha \in [1, 2)$. The proof requires analytic regularity results for the associated integro-differential operators of Kolmogorov type. We also study differentiability of solutions with respect to initial conditions and the homeomorphism property.

Generalised spin structures, or $r$-spin structures, on a $2$-dimensional orbifold $\Sigma$ are $r$-fold fibrewise connected coverings (also called $r$\textsuperscript{th} roots) of its unit tangent bundle $ST\Sigma$. We investigate such structures on hyperbolic orbifolds. The conditions on $r$ for such structures to exist are given. The action of the diffeomorphism group of $\Sigma$ on the set of $r$-spin structures is described, and we determine the number of orbits under this action and their size. These results are then applied to describe the moduli space of taut contact circles on left-quotients of the $3$-dimensional geometry $\widetilde{\mathrm{SL}}_{2}$.

We introduce a modified rack algebra $\mathbb{Z}[X]$ for racks $X$ with finite rack rank $N$. We use representations of $\mathbb{Z}[X]$ into rings, known as rack modules, to define enhancements of the rack counting invariant for classical and virtual knots and links. We provide computations and examples to show that the new invariants are strictly stronger than the unenhanced counting invariant and are not determined by the Jones or Alexander polynomials.

This paper deals with left invariant complex structures on simply connected Lie groups, the Lie algebra of which is of the type $\mathrm{T}_{\pi} \mathfrak{h}=\mathfrak{h} \ltimes_{\pi} V$, where $\pi$ is either the adjoint or the coadjoint representation. The main topic is the existence question of complex structures on $\mathrm{T}_{\pi} \mathfrak{h}$ for $\mathfrak{h}$ a three dimensional real Lie algebra. First it was proposed the study of complex structures $J$ satisfying the constraint $J\mathfrak{h} = V$. Whenever $\pi$ is the adjoint representation this kind of complex structures are associated to non-singular derivations of $\mathfrak{h}$. This fact allows different kinds of applications.

A Bott tower is an iterated $\mathbb{C}\mathrm{P}^{1}$-bundle over a point, where each $\mathbb{C}\mathrm{P}^{1}$-bundle is the projectivization of a rank $2$ decomposable complex vector bundle. For a Bott tower, the filtered cohomology is naturally defined. We show that isomorphism classes of Bott towers are distinguished by their filtered cohomology rings. We even show that any filtered cohomology ring isomorphism between two Bott towers is induced by an isomorphism of the Bott towers.

We show some results on the unknotting number and the band-unknotting number. Taniyama characterized knots whose unknotting number is half the crossing number minus one. We show that if the unknotting number of a knot is half the crossing number minus two, then the knot is the figure-eight knot, a positive $3$-braid knot, a negative $3$-braid knot or the connected sum of a $(2,r)$-torus knot and a $(2,s)$-torus knot for some odd integers $r,s \neq \pm 1$. In particular, we show that it is a $3$-braid knot. Taniyama and Yasuhara showed that the band-unknotting number of a knot is less than or equal to half the crossing number of the knot under our notation. We show that the equality holds if and only if the knot is the trivial knot or the figure-eight knot.